One of the first results one encounters in the standard university courses of algebra are classifications of equivalence classes of matrices under various transformations. If we are allowed to multiply the matrix in question from each side with an invertible matrix, then the corresponding equivalence class is fully determined by the rank of the matrix. If we consider equivalence classes of matrices under conjugations then two matrices are equivalent if and only if their normal Jordan forms are the same up to permutation of blocks.

I will explain how these questions can be putted to a more general framework of Representation Theory of finite dimensional algebras and acyclic quivers. It turns out that above two examples fall into two out of three distinct typical situation:

- there are only finitely many building blocks for canonical form (in the case of simultaneous transformations of columns and rows these blocks are 1-times-1 matrices (0) and (1) );

- in each dimension there are finitely many one-parameter families of building blocks for canonical form (in the case of conjugation of matrices these blocks are precisely the standard Jordan blocks);

- the classification of equivalence classes is unfeasible.

At the end of the talk I will give a short report on a recent joint work with K. Erdmann and A.P. Santana, which deals with representation type of Borel-Schur algebras.